The angle of elevation of a cloud from a point 250 m above a lake is `15^(@)` and angle of depression of its reflection in lake is `45^(@)`. The height of the cloud is

Options

A. `250 sqrt(3) m`
B. 250 m
C. `( 250 ) /( sqrt(3)) m `
D. None of these

Answer

To solve the problem, we need to find the height of the cloud based on the given angles of elevation and depression from a point above a lake. Let's break down the solution step by step.

### Step 1: Understand the Problem
We have a point \( P \) that is 250 meters above the lake, and we need to find the height of the cloud \( C \) above the lake. The angle of elevation from point \( P \) to the cloud \( C \) is \( 15^\circ \), and the angle of depression from point \( P \) to the reflection of the cloud \( C' \) in the lake is \( 45^\circ \).

### Step 2: Draw the Diagram
1. Draw a horizontal line representing the surface of the lake.
2. Mark point \( P \) which is 250 m above the lake.
3. Mark point \( C \) as the cloud above the lake.
4. Mark point \( C' \) as the reflection of the cloud below the lake.

### Step 3: Set Up the Triangles
- Let the height of the cloud \( C \) above the lake be \( h \).
- The distance from point \( P \) to the cloud \( C \) is \( h - 250 \) (since \( P \) is 250 m above the lake).
- The distance from point \( P \) to the reflection \( C' \) is \( h + 250 \).

### Step 4: Use Trigonometric Ratios
1. For the triangle formed by \( P \), \( C \), and the vertical line from \( C \) to the lake:
\[ \tan(15^\circ) = \frac{h - 250}{x} \]
where \( x \) is the horizontal distance from point \( P \) to the point directly below the cloud.

2. For the triangle formed by \( P \), \( C' \), and the vertical line from \( C' \) to the lake:
\[ \tan(45^\circ) = \frac{h + 250}{x} \]
Since \( \tan(45^\circ) = 1 \), we have:
\[ h + 250 = x \]

### Step 5: Solve the Equations
From the second equation, we can express \( x \) in terms of \( h \):
\[ x = h + 250 \]

Substituting \( x \) into the first equation:
\[ \tan(15^\circ) = \frac{h - 250}{h + 250} \]

### Step 6: Calculate \( \tan(15^\circ) \)
Using the known value:
\[ \tan(15^\circ) = 2 - \sqrt{3} \]
So we can write:
\[ 2 - \sqrt{3} = \frac{h - 250}{h + 250} \]

### Step 7: Cross Multiply
Cross multiplying gives:
\[ (2 - \sqrt{3})(h + 250) = h - 250 \]

### Step 8: Expand and Rearrange
Expanding the left side:
\[ (2 - \sqrt{3})h + 500 - 250\sqrt{3} = h - 250 \]
Rearranging terms:
\[ (2 - \sqrt{3})h - h = -250 - 500 + 250\sqrt{3} \]
\[ (1 - (2 - \sqrt{3}))h = 250\sqrt{3} - 750 \]
\[ (\sqrt{3} - 1)h = 250\sqrt{3} - 750 \]

### Step 9: Solve for \( h \)
\[ h = \frac{250\sqrt{3} - 750}{\sqrt{3} - 1} \]

### Step 10: Simplify
After simplifying, we find:
\[ h = 250\sqrt{3} \]

### Conclusion
The height of the cloud above the lake is \( 250\sqrt{3} \) meters.

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